Solving a novel designed second order nonlinear Lane-Emden delay differential model using the heuristic techniques

The aim of the present study is to present a new model based on the nonlinear singular second order delay differential equation of Lane–Emden type and numerically solved by using the heuristic technique. Four different examples are presented based on the designed model and numerically solved by using artificial neural networks optimized by the global search, local search methods and their hybrid combinations, respectively, named as genetic algorithm (GA), sequential quadratic programming (SQP) and GA-SQP. The numerical results of the designed model are compared for the proposed heuristic technique with the exact/explicit results that demonstrate the performance and correctness. Moreover, statistical investigations/assessments are presented for the accuracy and performance of the designed model implemented with heuristic methodology.This paper has been partially supported by Ministerio de Ciencia, Innovación y Universidades, Spain grant number PGC2018-0971-B-100 and Fundación Séneca de la Región de Murcia, Spain grant number 20783/PI/18

Evolutionary Integrated Heuristic with Gudermannian Neural Networks for Second Kind of Lane–Emden Nonlinear Singular Models

In this work, a new heuristic computing design is presented with an artificial intelligence approach to exploit the models with feed-forward (FF) Gudermannian neural networks (GNN) accomplished with global search capability of genetic algorithms (GA) combined with local convergence aptitude of active-set method (ASM), i.e., FF-GNN-GAASM to solve the second kind of Lane–Emden nonlinear singular models (LE-NSM). The proposed method based on the computing intelligent Gudermannian kernel is incorporated with the hidden layer configuration of FF-GNN models of differential operatives of the LE-NSM, which are arbitrarily associated with presenting an error-based objective function that is used to optimize by the hybrid heuristics of GAASM. Three LE-NSM-based examples are numerically solved to authenticate the effectiveness, accurateness, and efficiency of the suggested FF-GNN-GAASM. The reliability of the scheme via statistical valuations is verified in order to authenticate the stability, accuracy, and convergence

Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically

[EN]In this study, bio-inspired computational techniques have been exploited to get the numerical solution of a nonlinear two-point boundary value problem arising in the modelling of the corneal shape. The computational process of modelling and optimization makes enormously straightforward to obtain accurate approximate solutions of the corneal shape models through artificial neural networks, pattern search (PS), genetic algorithms (GAs), simulated annealing (SA), active-set technique (AST), interior-point technique, sequential quadratic programming and their hybrid forms based on GA–AST, PS–AST and SA–AST. Numerical results show that the designed solvers provide a reasonable precision and efficiency with minimal computational cost. The efficacy of the proposed computing strategies is also investigated through a descriptive statistical analysis by means of histogram illustrations, probability plots and one-way analysis of variance

Numerical Treatment of Non-Linear System for Latently Infected CD4+T Cells: A Swarm- Optimized Neural Network Approach

Swarm-inspired computing techniques are the best candidates for solving various nonlinear problems. The current study aims to exploit the swarm intelligence technique known as Particle Swarm Optimization (PSO) for the numerical investigation of a nonlinear system of latently infected CD4+T cells. The strength of the Mexican Hat Wavelet (MHW) based unsupervised Feed Forward Artificial Neural Network (FFANN) is used to solve the nonlinear system of latently infected CD4+T cells. The function approximation of unsupervised ANN is used to construct the mathematical model of the latently infected CD4+T cells by defining the error function in the mean square manner. The adjustable parameters called the unknowns of the network are optimized by using the Particle Swarm Optimization (PSO), Nedler Mead Simplex Method (NMSM), and their hybrid PSO-NMSM. The PSO applied for the global optimization of weights aided by the NMSM algorithm for rapid local search. Finally, a Comprehensive Monte Carlo simulation and statistical analysis of the analytical method, numerical Range Kutta (RK) method, ANN optimized with Genetic Algorithm (GA) aided with Sequential Quadratic Programming (SQP) known as GA-SQP, ANN-PSO-SQP and the proposed MHW-HIVFFANN-PSO-NMSM are performed to validate the effectiveness, stability, convergence, and computational complexity of each scheme. It is observed that the proposed MHW-FFANN-HIVPSO-NMSM scheme has converged in all classes at 10 −6 , 10−7 , and 10 −8 and solved the nonlinear system of latently infected CD4+ T cells more accurately and effectively. The absolute error lies in 10−3 , 10−4 , 10−4 , and 10−5 for numerical, ANN-GA-SQP, ANN-PSO-SQP, and proposed MHW-ANN-PSO-NMSM respectively. Moreover, the proposed scheme is stable for the large number of independent runs. The values for global statistical indicators’ global mean squared error are lies 8.15E-09, 3.25E-10, 4.15E-09, and 3.15E-10 for class X(t), W(t), Y(t), and V(t) respectively whereas the global mean absolute deviation lies in range 7.35E-09, 8.50E-10, 2.10E-10 and 7.10E-09

Design of neuro-computing paradigms for nonlinear nanofluidic systems of MHD Jeffery–Hamel flow

© 2018 Taiwan Institute of Chemical Engineers In this paper, a neuro-heuristic technique by incorporating artificial neural network models (NNMs) optimized with sequential quadratic programming (SQP) is proposed to solve the dynamics of nanofluidics system based on magneto-hydrodynamic (MHD) Jeffery–Hamel (JHF) problem involving nano-meterials. Original partial differential equations associated with MHD–JHF are transformed into third order ordinary differential equations based model. Furthermore, the transformed system has been implemented by the differential equation NNMs (DE-NNMs) which are constructed by a defined error function using log-sigmoid, radial basis and tan-sigmoid windowing kernels. The parameters of DE-NNM of nanofluidics system are optimized with SQP algorithm. To illustrate the performance of the proposed system, MHD–JHF models with base-fluid water mixed with alumina, silver and copper nanoparticles for different Hartman numbers, Reynolds numbers, angles of the channel and volume fractions with three different proposed DE-NNMs are designed to evaluate. For comparison purpose, the proposed results with reference numerical solutions of Adams solver illustrate their worth. Statistical inferences through different performance indices are given to demostrate the accuracy, stability and robustness of the stochastic solvers

Algebraic properties of ordinary differential equations.

Thesis (Ph.D.)-University of Natal, 1995.In Chapter One the theoretical basis for infinitesimal transformations is presented with particular emphasis on the central theme of this thesis which is the invariance of ordinary differential equations, and their first integrals, under infinitesimal transformations. The differential operators associated with these infinitesimal transformations constitute an algebra under the operation of taking the Lie Bracket. Some of the major results of Lie's work are recalled. The way to use the generators of symmetries to reduce the order of a differential equation and/or to find its first integrals is explained. The chapter concludes with a summary of the state of the art in the mid-seventies just before the work described here was initiated. Chapter Two describes the growing awareness of the algebraic properties of the paradigms of differential equations. This essentially ad hoc period demonstrated that there was value in studying the Lie method of extended groups for finding first integrals and so solutions of equations and systems of equations. This value was emphasised by the application of the method to a class of nonautonomous anharmonic equations which did not belong to the then pantheon of paradigms. The generalised Emden-Fowler equation provided a route to major development in the area of the theory of the conditions for the linearisation of second order equations. This was in addition to its own interest. The stage was now set to establish broad theoretical results and retreat from the particularism of the seventies. Chapters Three and Four deal with the linearisation theorems for second order equations and the classification of intrinsically nonlinear equations according to their algebras. The rather meagre results for systems of second order equations are recorded. In the fifth chapter the investigation is extended to higher order equations for which there are some major departures away from the pattern established at the second order level and reinforced by the central role played by these equations in a world still dominated by Newton. The classification of third order equations by their algebras is presented, but it must be admitted that the story of higher order equations is still very much incomplete. In the sixth chapter the relationships between first integrals and their algebras is explored for both first order integrals and those of higher orders. Again the peculiar position of second order equations is revealed. In the seventh chapter the generalised Emden-Fowler equation is given a more modern and complete treatment. The final chapter looks at one of the fundamental algebras associated with ordinary differential equations, the three element 8£(2, R), which is found in all higher order equations of maximal symmetry, is a fundamental feature of the Pinney equation which has played so prominent a role in the study of nonautonomous Hamiltonian systems in Physics and is the signature of Ermakov systems and their generalisations

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Testing general relativity with present and future astrophysical observations

One century after its formulation, Einstein's general relativity (GR) has made remarkable predictions and turned out to be compatible with all experimental tests. Most of these tests probe the theory in the weak-field regime, and there are theoretical and experimental reasons to believe that GR should be modified when gravitational fields are strong and spacetime curvature is large. The best astrophysical laboratories to probe strong-field gravity are black holes and neutron stars, whether isolated or in binary systems. We review the motivations to consider extensions of GR. We present a (necessarily incomplete) catalog of modified theories of gravity for which strong-field predictions have been computed and contrasted to Einstein's theory, and we summarize our current understanding of the structure and dynamics of compact objects in these theories. We discuss current bounds on modified gravity from binary pulsar and cosmological observations, and we highlight the potential of future gravitational wave measurements to inform us on the behavior of gravity in the strong-field regime